Restoring Cross-Step Feedback in Physics-Informed Neural Networks

Physics-Informed Neural Networks (PINNs) are mesh-free solvers for partial differential equations (PDEs), yet their training often suffers from two recurring difficulties. One is loss imbalance, where the gradients of the PDE residual and the boundary or initial losses can differ by orders of magnitude, complicating loss weighting. The other is poor internal propagation, where the backbone's layer-to-layer updates can make sharp or high-frequency solution features difficult to learn. Existing remedies largely address these issues separately, but adaptive weighting can remain sensitive to short-term fluctuations, while stronger backbones can still underfit difficult solution structures. These challenges motivate investigating decayed signal accumulation along both network depth and training time. We propose CSF-PINN (Cross-Step Feedback Physics-Informed Neural Network), which applies a common leaky-integral design at the architecture and loss-weighting levels. The Leaky-Integral Residual Block (LIRB) aggregates transformed features across depth through a recurrence equivalent to a damped hidden-state update with an initial-representation skip connection, while Integral-Controlled Weighting (ICW) combines instantaneous gradient statistics with a ratio of leaky integrals of square-root losses. Across five benchmark PDEs, CSF-PINN achieves the lowest error among all compared methods, including a 67.6% reduction in relative $L_2$ error over the strongest baseline on Wave.